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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Digital Gnosis

Frege’s philosophy of mathematics—Interview with Nathan Ormond, December 2021

Posted on October 10, 2021 by Joel David Hamkins
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I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.

The interview concludes with a public comment and question & answer session.

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Posted in Events, Talks, Videos | Tagged Digital Gnosis, Frege, philosophy of mathematics | Leave a reply

Infinitely More

The Hilbert program

An excerpt from Lectures on the Philosophy of Mathematics

Joel David Hamkins
7 HR AGO
14
Tactics versus strategies—the case of chess

Does chess admit of winning or drawing tactics? Which information exactly do we need to include as part of the board position?

Joel David Hamkins
Aug 17
8
6
The tactical variation of the fundamental theorem

We prove the tactical variation of the fundamental theorem of finite games—for finite games with sufficiently rich board positions, one of the players has a winning tactic or both have drawing tactics

Joel David Hamkins
Aug 10
6
2
Proof and the Art of Mathematics, MIT Press, 2020

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Recent Comments

  • Joseph Shipman on The elementary theory of surreal arithmetic is bi-interpretable with set theory, Kobe, Japan, September 2025
  • Joel David Hamkins on The elementary theory of surreal arithmetic is bi-interpretable with set theory, Kobe, Japan, September 2025
  • Joel David Hamkins on Largest-number contest: what is the largest number that you can describe on an index card?
  • G on Largest-number contest: what is the largest number that you can describe on an index card?
  • Joseph Shipman on The elementary theory of surreal arithmetic is bi-interpretable with set theory, Kobe, Japan, September 2025

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  • Comment by Joel David Hamkins on Which fields satisfy first-order induction?
    That's what I thought, and therefore also in the surreals and any RCF.
  • Comment by Joel David Hamkins on Which fields satisfy first-order induction?
    The natural notion of induction for the integers is a bi-directional induction, namely, if 𝐴 is a nonempty set of numbers and 𝑥 ∈𝐴 →𝑥 +1,𝑥 −1 ∈𝐴, then 𝐴 has all numbers. One could ask a version of your question for this bi-directional induction scheme. It's not true in ℚ, since ℤ is definable. But what […]
  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    Very nice, Andrej.
  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    Not always. Consider a diamond, where the max and min have two edges, or the one point order, where it has none.
  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    And when you refer to vertices in a partial order "with exactly one edge" are you referring instead to the minimal and maximal elements? Also, I normally think of posets as reflexive relations, but this would make all your digraphs have loops at every vertex. So I guess you want an irreflexive version of partial […]
  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    Could you clarify what you intend by the quotient of a partial order by an equivalence relation, when it is not a congruence? I guess you mean that every instance of the order relation suffices for an edge in the quotient?
  • Comment by Joel David Hamkins on Algorithms to count restricted injections
    Have you mixed up 𝑛 and 𝑚 in the beginning? You have 𝑓⁡(𝑎), where 𝑎 is from {1,…,𝑚}, but the domain of 𝑓 is said to be {1,…,𝑛}. If $n
  • Comment by Joel David Hamkins on Terminology: commonly used name for an 𝜔 machine?
    Of course ultimately the computational power has nothing to do with fitting the computation into finite time, but rather just the idea of making sense of a computation with infinitely many steps. So you may be interested in the Büchi automata (en.wikipedia.org/wiki/B%C3%BCchi_automaton), and beyond this, the infinite time Turing machines (jstor.org/stable/2586556), which extend the operation […]

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